googologywikiaorg-20200223-history
User blog:B1mb0w/The J Function
'REPLACED' This blog has been replaced. A new J Function blog should be referred to instead. The blog can be found here. 'The (old) J Function' The J function is a reasonably fast growing function. This blog will explain the structure of the function and will give various calculated values. The J Function will then be used by a brand new version of the Alpha function. Click here for more information about the Alpha Function. 'Introduction' For an introduction, it will be useful to refer to my other blogs on the Strong D function. The following notation rules will also help to understand the behaviour of Strong D Functions and the basic structure of the J Function. Notation: \(D(m_{x}) = D(m_1,m_2,...,m_x)\) \(D(m_{x},n_{y}) = D(m_1,m_2,...,m_x,n_1,n_2,...,n_y)\) \(D(1,0_{y}) = D(D(1_{y})_{y})\) \(D(m,n_{y}) = D(m-1,D(m,n_{y-1},n_y-1)_{y})\) Basic J Function Structure: The J Function consists of a series of 3-tuple parameters. This simple example uses one 3-tuple only: \(J(k,m,n) = D(k,0_{m},n)\) Note that the equivalent Strong D Function will have m+2 parameters in total, all parameters except for the first and last will be zero. Basic J Function Examples: \(J(2,0,3) = D(2,3)\) \(J(3,1,1) = D(3,0,1)\) \(J(4,2,0) = D(4,0,0,0)\) \(J(n,n-1,n) = D(n,0_{n-1},n) >> f_{\omega^2}(n)\) 'Definition' The J Function consists of a series of 3-tuple parameters. Each 3-tuple is simply 3 grouped input parameters. 3-tuples are delimited from each other with a semi-colon, instead of a comma. Here is a general example: \(J(1,2,3;4,5,6;7,8,9;...;...;...)\) J Function behaviour follows similar rules to the Strong D Function. The rules are as follows: \(J(k,m,n) = D(k,0_{m},n)\) \(J(k,m,n) = J(k-1,m,J(k,m,n-1))\) \(J(k,m,0) >> J(k-1,m,J(k-1,m,k))\) Explanation will be provided \(J(1,m,0) >> J(J(1,m-1,2^m-1),m-1,J(1,m-1,2^m-1))\) Explanation will be provided Note that: \(J(0,m,n) = J(0,0,n) = J(n) = D(n)\) Because of the Leading Zero rule: L1 and \(J(0,0,0) = J(0) = D(0) = 1\) When more than one 3-tuple is used we get more complex behaviour. The general rule is: \(J(f,g,h;k,m,n) = J(f,g,h;k-1,m,J(f,g,h;k,m,n-1))\) But it will be easier to follow the examples when f = g = 0 < h: \(J(h;k,m,n) = J(h;k-1,m,J(h;k,m,n-1))\) and then these rules follow from the definitions above. \(J(h;k,m,0) >> J(h;k-1,m,J(h;k-1,m,k))\) and \(J(h;1,m,0) >> J(h;J(h;1,m-1,2^m-1),m-1,J(h;1,m-1,2^m-1))\) Some simpler evaluations are: \(J(h;0,0,n) = J(h;n) = J(h;J(h;n-1))\) and \(J(h;0,0,0) = J(h-1;J(h-1;h,h,h),J(h-1;h,h,h),J(h-1;h,h,h))\) and \(J(1;0,0,0) = J(0;J(0;1,1,1),J(0;1,1,1),J(0;1,1,1))\) \(>> f_{\omega^2}(J(1,1,1)) = f_{\omega^2}(D(1,0,1)) >> f_{\omega^2}(f_3(6))\) 'Calculated Examples up to J(1;0,0,n)' \(J(1;0,0,1) = J(1;1) = J(1;J(1;0)) >> f_{\omega^2}(f_{\omega^2}(f_3(6))) >> f_{\omega^2}^2(f_3(6))\) and \(J(1;0,0,n-1) >> f_{\omega^2}^{n}(f_3(6)) >> f_{\omega^2+1}(n)\) when \(n < f_3(6)\) or \(J(1;0,0,n) >> f_{\omega^2+1}(n)\) when \(n < f_{\omega^2}(f_3(6))\) 'Calculated Examples up to J(1;n,0,n)' \(J(1;1,0,0) = J(1;0,0,J(1;0,0,1)) >> f_{\omega^2+1}(f_{\omega^2}^2(f_3(6)))\) and \(J(1;1,0,1) = J(1;0,0,J(1;1,0,0)) >> f_{\omega^2+1}^2(f_{\omega^2}^2(f_3(6)))\) and \(J(1;1,0,n-1) >> f_{\omega^2+1}^n(n) = f_{\omega^2+2}(n)\) when \(n < f_{\omega^2}^2(f_3(6))\) or \(J(1;1,0,n) >> f_{\omega^2+2}(n)\) when \(n < f_{\omega^2+1}(f_{\omega^2}^2(f_3(6)))\) then \(J(1;2,0,0) = J(1;1,0,J(1;1,0,2)) >> f_{\omega^2+2}(f_{\omega^2+1}^2(f_{\omega^2}^2(f_3(6))))\) and \(J(1;2,0,n) >> f_{\omega^2+3}(n)\) and \(J(1;3,0,n) >> f_{\omega^2+4}(n)\) then \(J(1;n,0,n) >> f_{\omega^2+\omega}(n)\) 'Calculated Examples up to J(1;n,n,n)' \(J(1;1,1,0) = J(1;J(1;1,0,1),0,J(1;1,0,1)) >> f_{\omega^2+\omega}(J(1;1,0,1))\) and \(J(1;1,1,1) = J(1;1,1,J(1;1,1,0)) >> f_{\omega^2+\omega}^2(J(1;1,0,1))\) then \(J(1;1,1,n) >> f_{\omega^2+\omega}^n(n) = f_{\omega^2+\omega+1}(n)\) and \(J(1;n,1,n) >> f_{\omega^2+\omega+n}(n)\) = f_{\omega^2+\omega.2}(n)\) then \(J(1;1,2,n) >> f_{\omega^2+\omega.2}^n(n) = f_{\omega^2+\omega.2+1}(n)\) and \(J(1;n,2,n) >> f_{\omega^2+\omega.2+n}(n)\) = f_{\omega^2+\omega.3}(n)\) then \(J(1;n,3,n) >> f_{\omega^2+\omega.4}(n)\) and \(J(1;n,n,n) >> f_{\omega^2+\omega.n}(n) >> f_{\omega^2.2}(n)\) 'Calculated Examples up to J(n;0,0,0)' \(J(2;0,0,0) = J(1;J(1;2,2,2),J(1;2,2,2),J(1;2,2,2)) >> f_{\omega^2.2}(J(1;2,2,2))\) and \(J(2;0,0,1) = J(2;0,0,J(2;0,0,0)) >> f_{\omega^2.2}^2(J(1;2,2,2))\) then \(J(2;0,0,n) >> f_{\omega^2.2+2}(n)\) and \(J(2;n,0,n) >> f_{\omega^2.2+\omega}(n)\) and \(J(2;n,n,n) >> f_{\omega^2.3}(n)\) or \(J(3;0,0,0) >> f_{\omega^2.3}(n)\) then \(J(n-1;n,n,n) >> f_{\omega^2.n}(n)\) or \(J(n;0,0,0) >> f_{\omega^3}(n)\) 'Calculated Examples up to J(1,0,n;n,n,n)' If we return to the general rule: \(J(f,g,h;k,m,n) = J(f,g,h;k-1,m,J(f,g,h;k,m,n-1))\) then \(J(1,0,0;k,m,n) = J(1,0,0;J(1,0,0;k,m,n-1),J(1,0,0;k,m,n-1),J(1,0,0;k,m,n-1))\) and \(J(1,0,0;0,0,0) = J(0,0,1;Z,Z,Z) = J(1;Z,Z,Z)\) where \(Z = J(0,0,1;D(1,0),D(1,0),D(1,0))\) Explanation will be provided or \(Z = J(1;D(1,0),D(1,0),D(1,0)) = J(1;3,3,3)\) then \(J(1,0,0;0,0,0) = J(1;J(1;3,3,3),J(1;3,3,3),J(1;3,3,3))\) \(>> f_{\omega^2.2}(J(1;3,3,3))\) and \(J(1,0,0;0,0,1) >> f_{\omega^2.2}^2(J(1;3,3,3))\) \(J(1,0,0;0,0,n) >> f_{\omega^2.2+1}(n)\) \(J(1,0,0;n,0,n) >> f_{\omega^2+\omega}(n)\) \(J(1,0,0;n,n,n) >> f_{\omega^3}(n)\) then \(J(1,0,1;0,0,0) >> J(1,0,0;J(1,0,0;4,4,4),J(1,0,0;4,4,4),J(1,0,0;4,4,4))\) Explanation will be provided \(>> f_{\omega^3}(f_{\omega^3}(4))\) and \(J(1,0,1;0,0,n) >> f_{\omega^3+1}(n)\) \(J(1,0,1;n,0,n) >> f_{\omega^3+\omega}(n)\) \(J(1,0,1;n,n,n) >> f_{\omega^3+\omega^2}(n)\) then \(J(1,0,2;0,0,n) >> f_{\omega^3+\omega^2+1}(n)\) \(J(1,0,2;n,0,n) >> f_{\omega^3+\omega^2+\omega}(n)\) \(J(1,0,2;n,n,n) >> f_{\omega^3+\omega^2.2}(n)\) then \(J(1,0,3;n,n,n) >> f_{\omega^3+\omega^2.3}(n)\) and \(J(1,0,n;n,n,n) >> f_{\omega^3.2}(n)\) 'Calculated Examples up to J(2,0,n;n,n,n)' Without proof, calculations should continue with: \(J(2,0,0;0,0,n) >> f_{\omega^3.2+1}(n)\) \(J(2,0,0;n,0,n) >> f_{\omega^3.2+\omega}(n)\) \(J(2,0,0;n,n,n) >> f_{\omega^3.2+\omega^2}(n)\) and \(J(2,0,1;n,n,n) >> f_{\omega^3.2+\omega^2.2}(n)\) \(J(2,0,2;n,n,n) >> f_{\omega^3.2+\omega^2.3}(n)\) \(J(2,0,h-1;n,n,n) >> f_{\omega^3.2+\omega^2.h}(n)\) and \(J(2,0,n;n,n,n) >> f_{\omega^3.3}(n)\) 'Calculated Examples up to J(n,n,n;n,n,n)' Without proof, calculations should continue with: \(J(3,0,n;n,n,n) >> f_{\omega^3.4}(n)\) \(J(f,0,n;n,n,n) >> f_{\omega^3.f}(n)\) and \(J(n,0,n;n,n,n) >> f_{\omega^4}(n)\) then \(J(1,1,0;0,0,n) >> f_{\omega^4+1}(n)\) \(J(1,1,0;n,0,n) >> f_{\omega^4+\omega}(n)\) \(J(1,1,0;n,n,n) >> f_{\omega^4+\omega^2}(n)\) \(J(1,1,n;n,n,n) >> f_{\omega^4+\omega^3}(n)\) \(J(n,1,n;n,n,n) >> f_{\omega^4.2}(n)\) \(J(n,g,n;n,n,n) >> f_{\omega^4.g}(n)\) and \(J(n,n,n;n,n,n) >> f_{\omega^5}(n)\) 'Growth Rate of the J Function' \(J(3;0,0,0) >> f_{\omega^3}(3) = f_{\omega^{\omega}}(3)\) \(J(4,0,4;4,4,4) >> f_{\omega^4}(n) = f_{\omega^{\omega}}(4)\) \(J(5,5,5;5,5,5) >> f_{\omega^5}(n) = f_{\omega^{\omega}}(5)\) The growth rate of the J Function therefore appears to be: \(J(n,n,n;n,n,n;...;n,n,n) >> f_{\omega^{\omega}}(n)\) with n+1 parameters in the J function Each group of 3 parameters of value n represents one 3-tuple in the J Function. If we use the notation N to represent the 3-tuple n,n,n, then the growth rate of the J function may reach: \(J(N_{(n+1)/3}) >> f_{\omega^{\omega}}(n)\) with n+1 parameters in the J function \(J(N_{n}) = J(N;N;...;N) = J(n,n,n;n,n,n;...;n,n,n) >> f_{\omega^{\omega^2}}(n)\) 'Some calculations for n=3' \(J(4,0,1) >> f_{\omega}(3)\) \(J(3,1,1) >> f_{\omega.2}(3)\) \(J(3,2,1) >> f_{\omega^2}(3)\) \(J(3;0,0,0) >> f_{\omega^{\omega}}(3)\) \(J(3;3,3,3;3,3,3) >> f_{\omega^{\omega.2}}(3)\) If we use the notation T to represent the 3-tuple 3,3,3, then we can continue: \(J(3;T;T;T) = J(3;3,3,3;3,3,3;3,3,3) >> f_{\omega^{\omega^2}}(3)\) 'Comments and Questions' Look forward to any comments and questions. If anybody is interested, the J Function was named by my wife. The full name is the Juki Function. Cheers B1mb0w. 'References' The J Function *Strong D Function Category:Blog posts